微积分学/曲面积分

例題

求 $x=e^t-t,y=4e^{\frac{t}{2}},0<t<1$ 繞 y 軸的表面積

$\begin{array}{cc}& A=\underset{C}{\int }2\pi rdS=\int 2\pi xd\sqrt{x^2+y^2}\\ & =\int 2\pi x\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt\\ & =\underset{0}{\overset{1}{\int }}2\pi x\sqrt{{(e^t-1)}^2+{\left(2e^{\frac{t}{2}}\right)}^2}dt\\ & =\underset{0}{\overset{1}{\int }}2\pi x\sqrt{e^{2t}-2e^t+1+4e^t}dt\\ & =\underset{0}{\overset{1}{\int }}2\pi x\sqrt{e^{2t}+2e^t+1}dt\\ & =\underset{0}{\overset{1}{\int }}2\pi (e^t-t)(e^t+1)dt\\ & =2\pi \underset{0}{\overset{1}{\int }}{e}^{2t}+e^t-t{e}^t-tdt\\ & =2\pi {[\frac{e^{2t}}{2}+e^t-(t{e}^t-e^t)-\frac{t^2}{2}]}_0^1\\ & =2\pi [\frac{e^2}{2}+e-\frac{1}{2}-(\frac{1}{2}+1+1)]=\pi (e^2+2e-6)\end{array}$